On the residual finiteness and other properties of (relative) one-relator groups

A relative one-relator presentation has the form $\mathcal{P} = \langle \mathbf{x}, H; R \rangle$ where $\mathbf{x}$ is a set, $H$ is a group, and $R$ is a word on $\mathbf{x}^{\pm 1} \cup H$. We show that if the word on $\mathbf{x}^{\pm 1}$ obtained from $R$ by deleting all the terms from $H$ has what we call the unique max-min property, then the group defined by $\mathcal{P}$ is residually finite if and only if $H$ is residually finite (Theorem 1). We apply this to obtain new results concerning the residual finiteness of (ordinary) one-relator groups (Theorem 4). We also obtain results concerning the conjugacy problem for one-relator groups (Theorem 5), and results concerning the relative asphericity of presentations of the form $\mathcal{P}$ (Theorem 6).